I am wondering about a question related to this post. According to bitcoinity, the number of bitcoin-trades at major exchanges per minute is roughly 400. However, as also pointed out by @eponymous in this answer, this number exceeds the trades which are verified on the bitcoin blockchain by far. However, how should I trust exchanges who do not provide real transactions in the sense that bitcoins are moved to my wallet? Is there a documentation on which exchanges provide this service and what exactly happens if I trade, say, USD versus BTC on a exchange which is not settling the trade on the blockchain?

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If you send 0.1 bitcoin to an exchange, you would see it credited to your account, but that doesn't mean that it is in a wallet solely dedicated to your account. Instead, that 0.1 bitcoin is sent to a wallet which aggregates all transactions to the exchange.

When you withdraw the bitcoin, a transaction is published from the aggregating wallet to your supplied bitcoin address.

All internal movements are just crediting and bookkeeping to the internal systems of the exchange. This is the reason why it is generally not advised to keep bitcoin an an exchange for very long. If the company goes bankrupt or is hacked, you won't be able to retrieve your funds, because you don't have control over the wallet.

As an example a sale (could be a buy or an exchange): You send some BTC to an exchange as a deposit to your account. The blockchain records that transaction. With respect to the blockchain the exchange owns those BTC. Now you sell the BTC via the exchange. This has no blockchain effect. The exchange does a bookkeeping entry to deduct BTC from your account and add them to the buyer's account. There will be no blockchain effect until the exchange sends those BTC (or part of them) to someone who withdraws them from the exchange after having bought them there or exchanged some altcoin for them.

As the @eponymous says, these are internal transactions which don't result in the actual moving of bitcoin. I would minimize the time your bitcoin sits on the exchanges. If it's not in a wallet only you control, it's not your bitcoin.

The Blockchain.info wallet allows you to buy bitcoin and transfers it on chain, but that's more of a coin selling platform. The high velocity exchanges wouldn't offer that service because they are geared for customers who want to trade quickly between different currencies.